Multi-Marginal Flow Matching with Adversarially Learnt Interpolants

TL;DR

ALI-CFM addresses the problem of inferring time-varying dynamics from multi-marginal snapshots by learning adversarial interpolants that match intermediate marginals and marginalising them with conditional flow matching to recover a consistent vector field. It introduces three regularisers to ensure unique, smooth interpolants and demonstrates strong performance on spatial transcriptomics, cell tracking, and single-cell trajectory inference, while remaining competitive in high-dimensional settings. The approach handles discrete and continuous time grids and shows robustness to noise and geometry changes over time, offering a scalable tool for interpreting complex biological dynamics. Overall, ALI-CFM advances multi-marginal trajectory inference by combining distributional interpolation via GANs with principled flow-based dynamics estimation, improving accuracy and stability in challenging data environments.

Abstract

Learning the dynamics of a process given sampled observations at several time points is an important but difficult task in many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at discrete time steps, the problem of modelling the dynamics, and thus inferring the underlying trajectories, can be solved by multi-marginal generalisations of flow matching algorithms. This paper proposes a novel flow matching method that overcomes the limitations of existing multi-marginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss to fit neurally parametrised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. We showcase the versatility and scalability of our method by outperforming the existing baselines on spatial transcriptomics and cell tracking datasets, while performing on par with them on single-cell trajectory prediction. Code: https://github.com/mmacosha/adversarially-learned-interpolants.

Figures (8)

• Figure 1: Our adversarially learnt interpolants (red curves) follow push-forward distributions (red densities) that approximate the intermediate-time marginal distributions $q_{t_j}$ and, by construction, have the correct end-marginals $q_0$ and $q_1$.
• Figure 2: Comparison of CFM tong2023improving, MFM kapusniak2024metric, MMFM rohbeck2025modeling and our ALI-CFM method on a synthetic 2D 'knot' distribution. See \ref{['sec:experiments']} for details.
• Figure 3: (a) and (b) show ten ALIs and piecewise linear interpolants, respectively, based on the subsampled cell tracking data, while (c) and (d) depict the resulting OT-ALI-CFM and OT-CFM vector fields. In the 2D figures we plot the centroids (one per frame) of (e) all of the segmentation data, (f) the subset training data, (g) the OT-ALI-CFM trajectories and (h) the OT-CFM trajectories. Note that the OT-CFM trajectories diverge in (d) and (h).
• Figure 4: From left to right we visualize frame 1, 25, 50, 75, 100 and 115 in the sequence of microscopy images. From top to bottom we overlay the images with the subsampled training data (10 samples per frame), and all segmentation samples $x_0$ pushed to the frame-specific times via the marginalised vector fields of OT-ALI-CFM, OT-CFM and OT-MFM. In concordance with the centroid plot in \ref{['fig:cell_centroids_ot_cfm']}, the trajectories from OT-CFM eventually diverge and become inaccurate. Meanwhile the time-independent metric in OT-MFM results in trajectories that do not capture the cell positions' temporal dependence.
• Figure 5: The aligned breast cancer H&E-stained images with overlaid scatter plots of tumor annotated coordinates (in light red).